Subgroups — C₂²×#360×C₆₇ #30712637

Order 17152.

1214 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#360×C₆₇	C₂	1	12	yes	maximal, direct product factor
5	C₂²×#41×C₆₇	—	2	1	no	—
6	C₂×#135×C₆₇	—	2	3	no	—
7	C₂×#135×C₆₇	—	2	3	no	—
8	C₂×#135×C₆₇	C₄	1	6	yes	complemented
9	C₂×#135×C₆₇	C₂²	1	6	yes	—
10	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic, complemented
11	C₂²×SD₁₆×C₆₇	—	2	1	no	—
12	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
15	C₂×M₃₂×C₆₇	—	2	3	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	6	yes	complemented
17	C₂²×C₁₆×C₆₇	—	2	1	no	—
18	C₂×#106×C₆₇	—	2	3	no	—
19	C₂×#106×C₆₇	C₂²	1	6	yes	complemented
20	#360×C₆₇	C₂²	1	16	yes	direct product factor
21	C₂×#41×C₆₇	—	2	6	no	—
22	C₂×#41×C₆₇	—	2	6	no	—
23	C₂×#41×C₆₇	—	4	3	no	—
24	C₂×#41×C₆₇	—	4	3	no	—
25	C₂²×Q₈×C₆₇	—	2	1	no	—
26	C₂²×D₄×C₆₇	—	2	1	no	—
27	C₂²×D₄×C₆₇	—	4	1	no	—
28	C₂³×C₄×C₆₇	—	4	1	no	—
29	C₂³×C₄×C₆₇	—	4	1	no	—
30	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
31	#135×C₆₇	—	2	12	no	—
32	#135×C₆₇	—	2	6	no	—
33	#135×C₆₇	—	2	6	no	—
34	#135×C₆₇	C₂×C₄	1	12	yes	complemented
35	#135×C₆₇	C₂³	1	4	yes	—
36	C₂×Q₁₆×C₆₇	—	2	3	no	—
37	C₂×Q₁₆×C₆₇	C₂×C₄	1	6	yes	complemented
38	C₂×SD₁₆×C₆₇	—	2	6	no	—
39	C₂×SD₁₆×C₆₇	—	2	6	no	—
40	C₂×D₈×C₆₇	—	2	3	no	—
41	C₂×D₈×C₆₇	C₂×C₄	1	6	yes	complemented
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₆₇	D₄	1	3	yes	—
45	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	D₄	1	3	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
49	C₂×C₄⋊C₄×C₆₇	—	4	3	no	—
50	M₃₂×C₆₇	—	2	6	no	—
51	M₃₂×C₆₇	C₂³	1	4	yes	complemented
52	C₂×C₁₆×C₆₇	—	2	6	no	—
53	#106×C₆₇	—	2	6	no	—
54	#106×C₆₇	C₂³	1	4	yes	complemented
55	C₂⁴×C₆₇	—	4	1	no	—
56	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
57	#41×C₆₇	—	2	12	no	—
58	#41×C₆₇	—	2	4	no	—
59	#41×C₆₇	—	4	12	no	—
60	#41×C₆₇	—	4	6	no	—
61	#41×C₆₇	—	4	6	no	—
62	C₂×Q₈×C₆₇	—	2	6	no	—
63	C₂×Q₈×C₆₇	—	4	3	no	—
64	C₂×D₄×C₆₇	—	2	6	no	—
65	C₂×D₄×C₆₇	—	4	6	no	—
66	C₂×D₄×C₆₇	—	4	6	no	—
67	C₂×D₄×C₆₇	—	4	3	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	6	no	—
70	C₂²×C₄×C₆₇	—	4	6	no	—
71	C₂²×C₄×C₆₇	—	4	6	no	—
72	C₂²×C₄×C₆₇	—	8	4	no	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	D₈	1	3	yes	—
76	C₂²×C₄×C₆₇	SD₁₆	1	3	yes	—
77	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
79	Q₁₆×C₆₇	—	2	6	no	—
80	Q₁₆×C₆₇	C₂²×C₄	1	4	yes	complemented
81	SD₁₆×C₆₇	—	2	12	no	—
82	SD₁₆×C₆₇	—	2	4	no	—
83	D₈×C₆₇	—	2	6	no	—
84	D₈×C₆₇	C₂²×C₄	1	4	yes	complemented
85	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	3	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	6	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
92	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
93	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	4	6	no	—
96	C₁₆×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	—	2	4	no	—
98	C₂³×C₆₇	—	4	6	no	—
99	C₂³×C₆₇	—	8	4	no	—
100	C₂³×C₆₇	#102	1	3	yes	—
101	C₂³×C₆₇	#102	1	1	yes	characteristic
102	C₂³×C₆₇	C₂×D₈	1	3	yes	—
103	Q₈×C₆₇	—	2	4	no	—
104	Q₈×C₆₇	—	4	6	no	—
105	D₄×C₆₇	—	2	4	no	—
106	D₄×C₆₇	—	4	4	no	—
107	D₄×C₆₇	—	4	12	no	—
108	D₄×C₆₇	—	4	6	no	—
109	C₂×C₄×C₆₇	—	4	6	no	—
110	C₂×C₄×C₆₇	—	4	4	no	—
111	C₂×C₄×C₆₇	—	4	12	no	—
112	C₂×C₄×C₆₇	—	4	4	no	—
113	C₂×C₄×C₆₇	—	8	12	no	—
114	C₂×C₄×C₆₇	#102	1	3	yes	—
115	C₂×C₄×C₆₇	#102	1	3	yes	—
116	C₂×C₄×C₆₇	#102	1	6	yes	—
117	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
118	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
119	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
120	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
121	C₂×C₄×C₆₇	C₂×SD₁₆	1	3	yes	—
122	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
124	C₈×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
125	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
126	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₆₇	—	2	6	no	—
128	C₂²×C₆₇	—	2	6	no	—
129	C₂²×C₆₇	—	4	4	no	—
130	C₂²×C₆₇	—	8	12	no	—
131	C₂²×C₆₇	#360	1	4	yes	direct product factor
132	C₂²×C₆₇	C₂×#102	1	3	yes	—
133	C₂²×C₆₇	C₂×#102	1	3	yes	—
134	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₆₇	—	4	4	no	—
136	C₄×C₆₇	—	8	4	no	—
137	C₄×C₆₇	C₂×#102	1	3	yes	—
138	C₄×C₆₇	C₂×#102	1	3	yes	—
139	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
141	C₂²×#360	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	—	2	3	no	—
144	C₂×C₆₇	—	8	4	no	—
145	C₂×C₆₇	C₂×#360	1	6	yes	direct product factor
146	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
147	C₂²×#135	C₂×C₆₇	1	1	yes	characteristic
148	C₂²×M₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
149	C₂²×#106	C₂×C₆₇	1	1	yes	characteristic, complemented
150	C₂×#360	C₂×C₆₇	1	12	yes	direct product factor
151	C₆₇	C₂²×#360	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×#41	—	2	1	no	—
153	C₂×#135	—	2	3	no	—
154	C₂×#135	—	2	3	no	—
155	C₂×#135	C₄×C₆₇	1	6	yes	complemented
156	C₂×#135	C₂²×C₆₇	1	6	yes	—
157	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic, complemented
158	C₂²×SD₁₆	—	2	1	no	—
159	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic, complemented
160	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
161	C₂²×C₄⋊C₄	—	2	1	no	—
162	C₂×M₃₂	—	2	3	no	—
163	C₂×M₃₂	C₂²×C₆₇	1	6	yes	complemented
164	C₂²×C₁₆	—	2	1	no	—
165	C₂×#106	—	2	3	no	—
166	C₂×#106	C₂²×C₆₇	1	6	yes	complemented
167	#360	C₂²×C₆₇	1	16	yes	direct product factor
168	C₂×#41	—	2	6	no	—
169	C₂×#41	—	2	6	no	—
170	C₂×#41	—	4	3	no	—
171	C₂×#41	—	4	3	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×D₄	—	2	1	no	—
174	C₂²×D₄	—	4	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
178	#135	—	2	12	no	—
179	#135	—	2	6	no	—
180	#135	—	2	6	no	—
181	#135	C₂×C₄×C₆₇	1	12	yes	complemented
182	#135	C₂³×C₆₇	1	4	yes	—
183	C₂×Q₁₆	—	2	3	no	—
184	C₂×Q₁₆	C₂×C₄×C₆₇	1	6	yes	complemented
185	C₂×SD₁₆	—	2	6	no	—
186	C₂×SD₁₆	—	2	6	no	—
187	C₂×D₈	—	2	3	no	—
188	C₂×D₈	C₂×C₄×C₆₇	1	6	yes	complemented
189	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂×C₄×C₆₇	1	3	yes	—
191	C₂²×C₈	D₄×C₆₇	1	3	yes	—
192	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₆₇	1	3	yes	—
194	C₂²×C₈	C₂³×C₆₇	1	3	yes	—
195	C₂×C₄⋊C₄	—	2	6	no	—
196	C₂×C₄⋊C₄	—	4	3	no	—
197	M₃₂	—	2	6	no	—
198	M₃₂	C₂³×C₆₇	1	4	yes	complemented
199	C₂×C₁₆	—	2	6	no	—
200	#106	—	2	6	no	—
201	#106	C₂³×C₆₇	1	4	yes	complemented
202	C₂⁴	—	4	1	no	—
203	C₂⁴	D₈×C₆₇	1	1	yes	characteristic
204	#41	—	2	12	no	—
205	#41	—	2	4	no	—
206	#41	—	4	12	no	—
207	#41	—	4	6	no	—
208	#41	—	4	6	no	—
209	C₂×Q₈	—	2	6	no	—
210	C₂×Q₈	—	4	3	no	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	4	6	no	—
213	C₂×D₄	—	4	6	no	—
214	C₂×D₄	—	4	3	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	6	no	—
217	C₂²×C₄	—	4	6	no	—
218	C₂²×C₄	—	4	6	no	—
219	C₂²×C₄	—	8	4	no	—
220	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊C₄×C₆₇	1	3	yes	—
222	C₂²×C₄	D₈×C₆₇	1	3	yes	—
223	C₂²×C₄	SD₁₆×C₆₇	1	1	yes	characteristic
224	C₂²×C₄	SD₁₆×C₆₇	1	3	yes	—
225	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
226	Q₁₆	—	2	6	no	—
227	Q₁₆	C₂²×C₄×C₆₇	1	4	yes	complemented
228	SD₁₆	—	2	12	no	—
229	SD₁₆	—	2	4	no	—
230	D₈	—	2	6	no	—
231	D₈	C₂²×C₄×C₆₇	1	4	yes	complemented
232	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
233	C₂×C₈	C₂²⋊C₄×C₆₇	1	3	yes	—
234	C₂×C₈	C₂²⋊C₄×C₆₇	1	6	yes	—
235	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
236	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
237	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
238	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
239	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
240	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	6	no	—
243	C₁₆	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	6	no	—
246	C₂³	—	8	4	no	—
247	C₂³	#102×C₆₇	1	3	yes	—
248	C₂³	#102×C₆₇	1	1	yes	characteristic
249	C₂³	C₂×D₈×C₆₇	1	3	yes	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	6	no	—
252	D₄	—	2	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	12	no	—
255	D₄	—	4	6	no	—
256	C₂×C₄	—	4	6	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	12	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	12	no	—
261	C₂×C₄	#102×C₆₇	1	3	yes	—
262	C₂×C₄	#102×C₆₇	1	3	yes	—
263	C₂×C₄	#102×C₆₇	1	6	yes	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
266	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
267	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
268	C₂×C₄	C₂×SD₁₆×C₆₇	1	3	yes	—
269	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
270	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
271	C₈	C₂×C₂²⋊C₄×C₆₇	1	3	yes	—
272	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
273	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
274	C₂²	—	2	6	no	—
275	C₂²	—	2	6	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	12	no	—
278	C₂²	#360×C₆₇	1	4	yes	direct product factor
279	C₂²	C₂×#102×C₆₇	1	3	yes	—
280	C₂²	C₂×#102×C₆₇	1	3	yes	—
281	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
282	C₄	—	4	4	no	—
283	C₄	—	8	4	no	—
284	C₄	C₂×#102×C₆₇	1	3	yes	—
285	C₄	C₂×#102×C₆₇	1	3	yes	—
286	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
287	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	3	no	—
290	C₂	—	8	4	no	—
291	C₂	C₂×#360×C₆₇	1	6	yes	direct product factor
292	C₂	C₂²×#102×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#102
commutator	C₈	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×#135×C₆₇	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×M₃₂×C₆₇	C₂	semidirect
3. C₂²×#106×C₆₇	C₂	semidirect
4. C₂×#360×C₆₇	C₂	direct (N × H)
8. C₂×#135×C₆₇	C₄	semidirect
10. C₂²×Q₁₆×C₆₇	C₄	semidirect
12. C₂²×D₈×C₆₇	C₄	semidirect
16. C₂×M₃₂×C₆₇	C₂²	semidirect
19. C₂×#106×C₆₇	C₂²	semidirect
20. #360×C₆₇	C₂²	direct (N × H)
34. #135×C₆₇	C₂×C₄	semidirect
37. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
41. C₂×D₈×C₆₇	C₂×C₄	semidirect
51. M₃₂×C₆₇	C₂³	semidirect
54. #106×C₆₇	C₂³	semidirect
80. Q₁₆×C₆₇	C₂²×C₄	semidirect
84. D₈×C₆₇	C₂²×C₄	semidirect
131. C₂²×C₆₇	#360	direct (N × H)
141. C₂²×#360	C₆₇	direct (N × H)
145. C₂×C₆₇	C₂×#360	direct (N × H)
148. C₂²×M₃₂	C₂×C₆₇	semidirect
149. C₂²×#106	C₂×C₆₇	semidirect
150. C₂×#360	C₂×C₆₇	direct (N × H)
151. C₆₇	C₂²×#360	direct (N × H)
155. C₂×#135	C₄×C₆₇	semidirect
157. C₂²×Q₁₆	C₄×C₆₇	semidirect
159. C₂²×D₈	C₄×C₆₇	semidirect
163. C₂×M₃₂	C₂²×C₆₇	semidirect
166. C₂×#106	C₂²×C₆₇	semidirect
167. #360	C₂²×C₆₇	direct (N × H)
181. #135	C₂×C₄×C₆₇	semidirect
184. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
188. C₂×D₈	C₂×C₄×C₆₇	semidirect
198. M₃₂	C₂³×C₆₇	semidirect
201. #106	C₂³×C₆₇	semidirect
227. Q₁₆	C₂²×C₄×C₆₇	semidirect
231. D₈	C₂²×C₄×C₆₇	semidirect
278. C₂²	#360×C₆₇	direct (N × H)
291. C₂	C₂×#360×C₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#135×C₆₇	2
C₂²×M₃₂×C₆₇	2
C₂²×#106×C₆₇	2
C₂×#360×C₆₇	2
C₂²×#360	67

Subgroups — C22×#360×C67 #30712637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#360×C₆₇ #30712637